Magnetic Fields
Magnets Domain = small magnetized region of a magnetic material all the atoms are grouped together and aligned
Magnets Ferromagnetic materials domains can be forced to line up by applying an eternal magnetic field permanent magnets: domains stay lined up after the eternal magnetic field is removed magnets always have both a North and a South pole Eamples: iron, cobalt, nickel
Magnetic Field B = magnetic field is a vector (has direction) points from a North magnetic pole to a South magnetic pole
Earth s Magnetic Field The earth s south magnetic pole is in the north The earth s north magnetic pole is in the south
Electromagnetism Magnetic fields eert a force on moving charged particles Electric currents (moving charged particles) create magnetic fields Aurora Borealis: Northern Lights
Force on a Charged Particle A magnetic field (B) eerts a magnetic force (FB) on a moving charged particle: FB qvb sin FB = magnetic force (in N) q = charge (in C) v = velocity (in m/s) B = magnetic field (in T = Tesla = N ) A m θ = angle between the direction of v and the direction of B
Magnetic Field Problem A proton travels at 1 105 m/s perpendicular to Earth s magnetic field of 55 μt Calculate the magnetic force (FB) on the proton FB qvb sin 19 6 (16 10 C )(1 10 m / s )(55 10 T ) sin( 90 ) 88 10 19 N 5
Direction Find the direction of the force using the right-hand-rule Point the inde finger of your right hand in the direction of v Point your middle finger in the direction of B Your thumb gives the direction of F Note: the direction is opposite for negative charges
Magnetic Field Problem Same problem: proton moving in magnetic field v = 1 105 m/s eastward B = 55 μt north Find the direction of the force Upward
Drawing Magnetic Fields means directed into the page (away from you) v Find the direction of FB means pointing out of the page (towards you) v
Force on a Moving Particle A (magnetic) force acts on a charged particle that moves in a magnetic field F B qv B cross product F B q v B sin
Direction of Force The direction of the magnetic force is determined by the right-hand-rule http://hyperphysicsphyastrgsuedu/hbase/magnetic/magforhtml#c3 The force on a charged particle moving through a magnetic field is always perpendicular to both v and B
Direction of Force Determine the direction of the magnetic force acting on a positively charged particle moving in a magnetic field: y y y B B v z B z v z v F=0
Magnetic Field Units T = Tesla = SI unit for magnetic field N N N 1T 1 1 1 C m / s (C / s )( m ) A m 1 G = 1 gauss (non-si unit) 1 T = 104 G
Circular Motion Problem An electron (mass m, charge q) moves in a magnetic field B at speed v, perpendicular to the magnetic field Determine the radius of curvature of the trajectory in terms of m, q, B, and v 2 v F m qvb r mv r qb
Magnetic Force Problem A beam of electrons eits an accelerator tube with kinetic energy K If it is a distance d from a metal plate, show that it will miss the plate if there is an applied magnetic field of 2mK B 2 2 ed electron beam accelerator tube d
Force on a Current Magnetic fields eert a force on a currentcarrying wire (charges are moving) F Id l B F = force cross product I = current dl = element of length B = magnetic field Simplifies to FB IlB sin θ
Direction of FB Right-hand-rule: Point inde finger of right hand in the direction of the current I Point your middle finger in the direction of B Thumb points to FB F Id l B
FB on a Current A wire carries a current of 22 A from west to east, at a point where the magnetic field is 50 10-5 T and points horizontally from south to north Find the magnitude and direction of the magnetic force on a 36 m length of wire
FB on a Current I = 22 A B = 5 10-5 T l = 36 m θ = 90 I FB IlB sin 5 (22 A)(36m)(5 10 T ) sin 90 upwards 00396 N
Magnetic Field due to a Current Moving charge (a current) produces a magnetic field Biot-Savart law: o Id l r db 3 4 r μo = 4π 10-7 T m/a = permeability constant B curls around I, according to the right-hand-rule
Long Straight Wire Calculate the magnetic field due to a long straight wire I o Id l r B db 3 r 4 θ dl o Idl r sin 3 r 4 o I 4 r sin dl r 2 R l
Long Straight Wire o I B 4 o I 4 o I 4 sin dl r 2 sin ( R / r )dl r 2 R r R I R2 l2 θ Rdl r 3 o IR dl 4 l 2 R 2 3 / 2 r o IR l 4 R 2 (l 2 R 2 )1/ 2 R dl l
integral Show that this is the integral by taking its derivative: o IR l B 4 R 2 (l 2 R 2 )1/ 2 d l 2 2 2 1/ 2 dl R (l R )
Long Straight Wire o IR l B 4 R 2 (l 2 R 2 )1/ 2 I o IR l 2 2 2 1/ 2 4 R (l R ) o I 1 ( 1) 4 R o I B 2 R θ r R dl l
Circular Arc of Wire Calculate the magnetic field at the center of an arc of wire of angle φ: o Id l r B db r3 4 o Idl r sin r3 4 o I dl 2 4 R R φ r dl
Circular Arc of Wire o I B dl 2 4 R o I Rd 2 4 R o IR d 2 4 R 0 o I B 4 R R φ r
Magnetic Field Rank the following circuits according to the magnetic field produced at the center of curvature (the dot) The circuits carry the same current greatest least
Torque on a Current Loop A loop of current in a magnetic field feels a torque due to the magnetic force: aligns the loop with the magnetic field Application: electric motors!
Parallel Currents Two parallel wires carry currents in the same direction Is there a force between them? If so, is it attractive or repulsive? d IA IB
Parallel Wires Parallel currents attract Antiparallel currents repel
Ampere s Law Gauss Law: find the electric field due to a distribution of charges (enclosed) qenc E da o Ampere s Law: find the magnetic field due to a distribution of currents (enclosed) B d l I o
Ampere s Law Find the magnetic field outside a long straight wire that carries a current I: B d l o I Integrate around loop B d l B cos dl B dl B(2 r ) B ( 2 r ) o I o I B 2 r Center = wire r B
Ampere s Law Find the magnetic field inside a solenoid of n loops per unit length http://hyperphysicsphy-astrgsuedu/hbase/magnetic/solenoidhtml
Solenoids Integrate around a rectangle that encircles the current loops on one side of the solenoid B dl I B = magnetic field l = length of solenoid Bl I = current in wire B N = number of coils n = number of coil per unit length μo = 4π 10-7 T m/a o o ( NI ) o In
Solenoid Problem A 200-turn solenoid having a length of 25 cm and a diameter of 10 cm carries a current of 030 A Calculate the magnitude of the magnetic field B inside the solenoid B = 3 10-4 T
Toroid A toroid is a solenoid in the shape of a donut: Use Ampere s Law to determine the magnetic field inside a toroid B dl I o B(2 r ) o ( NI ) o IN B 2 r
Magnetic Flu Magnetic flu = amount of magnetic field that passes through a given area φm = magnetic flu m B d A Units = Wb = T m2 Wb = Weber Note use of the dot product (use cosθ or parallel components of B and A)
Magnetic Flu link http://phetcoloradoedu/new/simulations/sims php?sim=faradays_law
Faraday s Law of Induction An emf is induced in a loop when the flu (number of magnetic field lines that pass through the loop) is changing: d m E dt
Faraday s Law link http://phetcoloradoedu/new/simulations/sims php?sim=faradays_electromagnetic_lab
Induced Emf Rank the regions according to the magnitude of the emf induced in a loop of wire B 1 3 2 2 3 t
Faraday Law Problem A long solenoid of 220 turns/cm and diameter 32 cm carries a current of 15A Its current is reduced to zero at a steady rate in 25 ms What is the magnitude of the emf induced in a coil of diameter 21 cm and 130 turns while the current in the solenoid is changing?
Faraday Law Problem To find emf need change in flu: m d m N E N for N turns t dt To find flu need magnetic field: m B d A BA cos for constant B
Faraday Law Problem Solenoid: n = 220 turns/cm = 22 105 turns/m I = 15 A d = 32 cm; r = 16 cm t = 25 ms = 25 10-3 m First find magnetic field B inside a solenoid: B o In Bi ( 4 10 7 T m / A)(15 A)( 22 10 4 tuns / m ) 00415T Bf 0
Faraday Law Problem Now find magnetic flu inside the coil, then emf: d = 21 cm; r = 105 cm = 105 10-2 m N = 130 turns m BA cos i (00415T )( (105 10 2 m ) 2 ) cos 0 144 10 5 Wb f 0 0 144 10 5 Wb m 00749V 75mV E N (130turns ) 3 25 10 s t
Lenz s Law Which way does the induced current flow? Lenz s Law: An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flu that induces the current opposes the change in flu
Lenz s Law Which direction is the induced current? Increasing B Counterclockwise Decreasing B Clockwise
Lenz s Law Which direction is the induced current? Increasing B Clockwise Decreasing B Counterclockwise
B field problem A rectangular loop of wire lies in a uniform magnetic field directed out of the page of magnitude B = 4t2 + 2t + 3 The loop is connected to a battery with emf Ebat = 20V 2 If an area of 01 m lies in the magnetic field, what is the induced emf in the loop at t = 10s?
B field problem B = 4t2 + 2t + 3 Ebat = 20V d m E dt d BA dt d (01)( 4t 2 2t 3 dt (01)(80 2) 82V A = 01 m2 t = 10 s m B d A BA cos 0 BA
B field problem What is the direction of the induced current? Magnetic field is increasing out of the page Current opposes increase Induced current is clockwise
B field problem Same problem: B = 4t2 + 2t + 3 Ebat = 20V induced emf = 82V induced I = clockwise A = 01 m2 t = 10 s If the conducting loop has a resistance of 10 Ω, what is the total current in the loop at 10 s? Enet 8 2V 2 0V 0 62 A I R 10
Inductors Inductors are little solenoids An induced emf EL appears in any coil (an inductor) in which the current is changing di E L dt self - induced emf L = inductance of inductor unit = henry = H = T m2/a L is a measure of the opposition to the rate of change of current
Inductance Inductors oppose change in a circuit I decreasing I increasing E E Initially, an inductor acts to oppose changes in the current through it A long time later, it acts like ordinary connecting wire
Uses for Inductors Used etensively in analog circuits and signal processing for tuning frequencies, filtering out specific frequencies 2 or more inductors with coupled magnetic flu form a transformer Large, heavy components; used less in modern equipment
RL Circuits Close circuit, start current: I Io 1 e t / L Open circuit, stop current: I I oe t / L Inductive time constant: L L R
RL Circuit An RL circuit is hooked up, as shown Plot the potential across the resistor and across the inductor as a function of time: VR VL t t
RL Circuits What is the current through the battery just after the switch is closed? Battery has E = 18V Resistors are all R = 9 Ω Inductors are all L = 2 mh E 18V I 2A R 9
RL Circuits What is the current through the battery after a long time? 1 1 1 1 Battery has E = 18V Req R1 R2 R3 Resistors are all R = 9 Ω Inductors are all L = 2 mh 1 1 1 1 Req 9 9 9 1 Req 3 3 E 18V 6A I R 3
RL Circuits Rank the circuits according to the current through the battery a) just after the switch is closed and b) a long time later
Inductors Energy is stored in an inductor: 1 2 U L LI 2
RL Circuit Problem I I 1 e A solenoid has an inductance of 53 mh I I 1 e t / L o 1 2 t / L o o and a resistance of 1 t / e 037 Ω 2 If it is connected to a t 1 ln ln 1 ln 2 battery, how long L 2 will the current take t L ln 2 to reach half its 53 10 3 H L final equilibrium t ln 2 ln 2 value? 037 R 010 s L
RL Circuit Problem Same circuit: L=53mH R=037Ω If the circuit is attached to a 12V battery, how much energy is stored after 010s (at half the equilibrium current)? 1 2 LI 2 I V / R (12V ) /( 037 ) 324 A UL 2 1 324 A 3 U L (53 10 H ) 697 J 2 2
LC Circuit First charge the capacitor with the switch at a, then throw the switch to b a C b L Describe the subsequent current in the circuit It oscillates!
LC circuit Write (but do not solve) the equation for the charge and/or current in the LC circuit when the switch is at b a C b L VC VL 0 q di L 0 C dt q d dq L 0 C dt dt d 2q q 0 2 dt LC T 2 LC